§16.21 Differential Equation

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Keywords:: Meijer -function, differential equation
Notes:: See Luke (1969a, Chapter 5).
Permalink:: http://dlmf.nist.gov/16.21
See also:: Annotations for Ch.16

$w={G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)$ satisfies the differential equation

16.21.1 $\left((-1)^{p-m-n}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b_{% 1})\cdots(\vartheta-b_{q})\right)w=0,$

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Symbols:: : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, : solution and $\vartheta$ : differential operator
Referenced by:: §16.21, §16.21
Permalink:: http://dlmf.nist.gov/16.21.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §16.21 and Ch.16

where again $\vartheta=z\ifrac{\mathrm{d}}{\mathrm{d}z}$ . This equation is of order $\max(p,q)$ . In consequence of (16.19.1) we may assume, without loss of generality, that $p\leq q$ . With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at and an irregular singularity at $z=\infty$ . When the only singularities of (16.21.1) are regular singularities at , $(-1)^{p-m-n}$ , and $\infty$ .

A fundamental set of solutions of (16.21.1) is given by

16.21.2 ${G^{1,p}_{p,q}}\left(z{\mathrm{e}}^{(p-m-n-1)\pi\mathrm{i}};{a_{1},\dots,a_{p}% \atop b_{j},b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\right),$ $j=1,\dots,q$ .

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Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.21.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §16.21 and Ch.16

For other fundamental sets see Erdélyi et al. (1953a, §5.4) and Marichev (1984).